Given $\alpha\in S$ , the proper initial segments of $\mathcal{E}_{\neg}(\alpha)=(\neg\alpha)$ are $($ , $(\neg$ , $(\neg\alpha_{0}$ (where $\alpha_{0}$ is a proper initial segment of $\alpha$ if one exists), $(\neg\alpha$ . Given that by the induction hypothesis $\alpha_{0}$ has more left parentheses than right ones, and that $\alpha$ has a balance of left and right parentheses, we conclude that all proper initial segments of $\mathcal{E}_{\neg}(\alpha)$ have more left parentheses than right ones.